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Analysis of experimental data: The average shape of extreme wave forces on monopilefoundations and the NewForce model

Schløer, Signe; Bredmose, Henrik; Ghadirian, Amin

Published in:Energy Procedia

Link to article, DOI:10.1016/j.egypro.2017.10.376

Publication date:2017

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Citation (APA):Schløer, S., Bredmose, H., & Ghadirian, A. (2017). Analysis of experimental data: The average shape ofextreme wave forces on monopile foundations and the NewForce model. Energy Procedia, 137, 223-237.https://doi.org/10.1016/j.egypro.2017.10.376

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1876-6102 © 2017 The Authors. Published by Elsevier Ltd.Peer-review under responsibility of the Scientific Committee of The 15th International Symposium on District Heating and Cooling.

The 15th International Symposium on District Heating and Cooling

Assessing the feasibility of using the heat demand-outdoor temperature function for a long-term district heat demand forecast

I. Andrića,b,c*, A. Pinaa, P. Ferrãoa, J. Fournierb., B. Lacarrièrec, O. Le Correc

aIN+ Center for Innovation, Technology and Policy Research - Instituto Superior Técnico, Av. Rovisco Pais 1, 1049-001 Lisbon, PortugalbVeolia Recherche & Innovation, 291 Avenue Dreyfous Daniel, 78520 Limay, France

cDépartement Systèmes Énergétiques et Environnement - IMT Atlantique, 4 rue Alfred Kastler, 44300 Nantes, France

Abstract

District heating networks are commonly addressed in the literature as one of the most effective solutions for decreasing the greenhouse gas emissions from the building sector. These systems require high investments which are returned through the heatsales. Due to the changed climate conditions and building renovation policies, heat demand in the future could decrease, prolonging the investment return period. The main scope of this paper is to assess the feasibility of using the heat demand – outdoor temperature function for heat demand forecast. The district of Alvalade, located in Lisbon (Portugal), was used as a case study. The district is consisted of 665 buildings that vary in both construction period and typology. Three weather scenarios (low, medium, high) and three district renovation scenarios were developed (shallow, intermediate, deep). To estimate the error, obtained heat demand values were compared with results from a dynamic heat demand model, previously developed and validated by the authors.The results showed that when only weather change is considered, the margin of error could be acceptable for some applications(the error in annual demand was lower than 20% for all weather scenarios considered). However, after introducing renovation scenarios, the error value increased up to 59.5% (depending on the weather and renovation scenarios combination considered). The value of slope coefficient increased on average within the range of 3.8% up to 8% per decade, that corresponds to the decrease in the number of heating hours of 22-139h during the heating season (depending on the combination of weather and renovation scenarios considered). On the other hand, function intercept increased for 7.8-12.7% per decade (depending on the coupled scenarios). The values suggested could be used to modify the function parameters for the scenarios considered, and improve the accuracy of heat demand estimations.

© 2017 The Authors. Published by Elsevier Ltd.Peer-review under responsibility of the Scientific Committee of The 15th International Symposium on District Heating and Cooling.

Keywords: Heat demand; Forecast; Climate change

Energy Procedia 137 (2017) 223–237

1876-6102 © 2017 The Authors. Published by Elsevier Ltd.Peer-review under responsibility of SINTEF Energi AS.10.1016/j.egypro.2017.10.376

10.1016/j.egypro.2017.10.376 1876-6102

Available online at www.sciencedirect.com

Energy Procedia 00 (2016) 000–000www.elsevier.com/locate/procedia

14th Deep Sea Offshore Wind R&D Conference, EERA DeepWind’2017, 18-20 January 2017,Trondheim, Norway

Analysis of experimental data: The average shape of extreme waveforces on monopile foundations and the NewForce model

Signe Schløera,∗, Henrik Bredmosea, Amin Ghadiriana

aDTU Wind Energy, Nils Koppels Alle, Building 403, DK-2800 Kgs. Lyngby, Denmark

Abstract

Experiments with a stiff pile subjected to extreme wave forces typical of offshore wind farm storm conditions are considered. Theexceedance probability curves of the nondimensional force peaks and crest heights are analysed. The average force time historynormalised with their peak values are compared across the sea states. It is found that the force shapes show a clear similarity whengrouped after the values of the normalised peak force, F/(ρghR2), normalised depth h/(gT 2

p) and presented in a normalised timescale t/Ta. For the largest force events, slamming can be seen as a distinct ’hat’ on top of the smoother underlying force curve.

The force shapes are numerically reproduced using a design force model, NewForce, which is introduced here for the first timeto both first and second order in wave steepness. For force shapes which are not asymmetric, the NewForce model compares wellto the average shapes. For more nonlinear wave shapes, higher order terms has to be considered in order for the NewForce modelto be able to predict the expected shapes.c� 2016 The Authors. Published by Elsevier Ltd.Peer-review under responsibility of SINTEF Energi AS.

Keywords: NewForce; Experimental data; Average force shape; Extreme forces

1. Introduction

In today’s design of offshore wind turbines, the extreme waves are based on probability distributions from mea-surements and hindcast models. In the dynamic analysis, the extreme waves are often calculated by stream functionwave theory embedded into a linear irregular time series. The wave period of the stream function wave should ac-cording to [9] be the period inside a predefined interval as function of the wave height, which gives the largest load.The wave force is typically calculated by Morison’s equation. Instead of stream function waves, NewWave theory,[1,11,17], can also be used to calculate the extreme waves. The NewWave is a deterministic design wave, where thecrest amplitude is predefined and the time history of the crest is described by the autocorrelation function of the wavespectrum, assuming the surface elevation to be a linear random Gaussian process.

The NewWave has been validated to experimental data and real measurements in a number of studies, cf. [3,8,15,18,19]. Whittaker et al., [19], showed that NewWave theory compares well to the linearised average shape of the

∗ Corresponding author. Tel.: +0-000-000-0000 ; fax: +0-000-000-0000.E-mail address: sigs@dtu.dk

1876-6102 c� 2016 The Authors. Published by Elsevier Ltd.Peer-review under responsibility of SINTEF Energi AS.

Available online at www.sciencedirect.com

Energy Procedia 00 (2016) 000–000www.elsevier.com/locate/procedia

14th Deep Sea Offshore Wind R&D Conference, EERA DeepWind’2017, 18-20 January 2017,Trondheim, Norway

Analysis of experimental data: The average shape of extreme waveforces on monopile foundations and the NewForce model

Signe Schløera,∗, Henrik Bredmosea, Amin Ghadiriana

aDTU Wind Energy, Nils Koppels Alle, Building 403, DK-2800 Kgs. Lyngby, Denmark

Abstract

Experiments with a stiff pile subjected to extreme wave forces typical of offshore wind farm storm conditions are considered. Theexceedance probability curves of the nondimensional force peaks and crest heights are analysed. The average force time historynormalised with their peak values are compared across the sea states. It is found that the force shapes show a clear similarity whengrouped after the values of the normalised peak force, F/(ρghR2), normalised depth h/(gT 2

p) and presented in a normalised timescale t/Ta. For the largest force events, slamming can be seen as a distinct ’hat’ on top of the smoother underlying force curve.

The force shapes are numerically reproduced using a design force model, NewForce, which is introduced here for the first timeto both first and second order in wave steepness. For force shapes which are not asymmetric, the NewForce model compares wellto the average shapes. For more nonlinear wave shapes, higher order terms has to be considered in order for the NewForce modelto be able to predict the expected shapes.c� 2016 The Authors. Published by Elsevier Ltd.Peer-review under responsibility of SINTEF Energi AS.

Keywords: NewForce; Experimental data; Average force shape; Extreme forces

1. Introduction

In today’s design of offshore wind turbines, the extreme waves are based on probability distributions from mea-surements and hindcast models. In the dynamic analysis, the extreme waves are often calculated by stream functionwave theory embedded into a linear irregular time series. The wave period of the stream function wave should ac-cording to [9] be the period inside a predefined interval as function of the wave height, which gives the largest load.The wave force is typically calculated by Morison’s equation. Instead of stream function waves, NewWave theory,[1,11,17], can also be used to calculate the extreme waves. The NewWave is a deterministic design wave, where thecrest amplitude is predefined and the time history of the crest is described by the autocorrelation function of the wavespectrum, assuming the surface elevation to be a linear random Gaussian process.

The NewWave has been validated to experimental data and real measurements in a number of studies, cf. [3,8,15,18,19]. Whittaker et al., [19], showed that NewWave theory compares well to the linearised average shape of the

∗ Corresponding author. Tel.: +0-000-000-0000 ; fax: +0-000-000-0000.E-mail address: sigs@dtu.dk

1876-6102 c� 2016 The Authors. Published by Elsevier Ltd.Peer-review under responsibility of SINTEF Energi AS.

© 2017 The Authors. Published by Elsevier Ltd.Peer-review under responsibility of SINTEF Energi AS.

224 Signe Schløer et al. / Energy Procedia 137 (2017) 223–2372 Author name / Energy Procedia 00 (2016) 000–000

largest waves during a storm for nondimensional wave numbers down to kh = 0.4, with k being the wave number andh the water depth. Further, a second-order corrected NewWave profile compared resonable to the average measuredwave profile.

However, as will be shown in this paper, it is not always the wave with the largest wave height or crest height,which results in the largest wave force. Instead, the largest force could be due to a wave with a smaller wave heightand amplitude but with a very steep wave crest. Less uncertainty on the extreme wave force would thus be obtained ifbased on probability distributions of the force itself instead of wave height.

For nonlinear regular waves, Paulsen et al. [12] found a similarity in the temporal development of the normalisedforce time histories, F/Fmax as function of H/Hmax, where H is the wave height and Hmax is the maximum waveheight at the given nondimensional depth from the wave breaking criterion of [20]. Further, the force peak value,Fmax/(ρghR2) was found to increase with H/Hmax but was independent of the nondimenisonal water depth kh. Here ρis the density of water, g the gravity and R the radius of the monopile.

In the present paper it is investigated if a similar correlation can be found for experimental measured extremeresponse forces on a monopile foundation from irregular waves, and whether these forces can be predicted numericallyif only information about the water depths, the significant wave heights, Hs and peak wave periods, Tp, of the waverealizations are known. To do this the NewForce model is introduced here for the first time. It can be seen as a designforce in same way as the NewWave theory for the free surface elevation. The model is compared to second orderFORM analysis in [6].

The paper opens with an outset of the analysis and a presentation of the NewForce model. Next the model testconsidered in the analysis are introduced and exceedance probability curves of the nondimensional crest heights ofthe free surface elevation and force peaks are presented for the different sea states. Based on the probability curves,the forces shapes conditional to a given normalised force peak level are compared across a the different sea states anda relation between the shapes are found. The force shapes are further compared to the NewForce model. The paperends with a summary and discussion.

2. Outset of the analysis

In Offshore Engineering, design wave loads are often derived with basis in a certain design wave height andsubsequent application of a wave theory and a force model. However, as the peak force is not just a function of waveheight but also depends on e.g. the wave front slope, the resulting design load is associated with uncertainty. Some ofthis uncertainty can be by-passed if the force statistics of the given structure is known. In that case, the design loadsfor a static design are readily available. However, in order to run dynamic calculations, information about the forcevariation in space and time is also necessary. The present paper seeks to provide these two pieces of information byanalysis of experimental measurements on a monopile. We will use the word shape as an alternative denotation oftime history throughout.

Consider a monopile of radius R at depth h, subjected to a JONSWAP sea state with significant wave height Hs,peak period Tp and peak enhancement factor γ. We further denote the water density by ρ, gravity by g and the waterviscosity by ν. A given realization of the sea state can be characterized by a set of stochastic variables {x j}, and forsuch a realization the force time series can be expressed by

FρghR2 = f

Hs

gT 2p,

hgT 2

p, γ,

RgT 2

p,ν

�

ghR2,

tTp, {x j}

. (1)

The force peaks may be sorted in increasing order to estimate the associated probability distribution, which may beexpressed as a function of the same parameters

P�

Fi

ρghR2 ≤FρghR2

�

= f1

Fi

ρghR2 ,Hs

gT 2p,

hgT 2

p, γ,

RgT 2

p,ν

�

ghR2

, (2)

where the time and stochastic variables {x j} are now removed.

Signe Schløer et al. / Energy Procedia 137 (2017) 223–237 225Author name / Energy Procedia 00 (2016) 000–000 3

For a given value of Fi, the expected force history can be determined by ensemble averaging. This expected timehistory can thus be expressed by

FFi= f2

Fi

ρghR2 ,Hs

gT 2p,

hgT 2

p, γ,

RgT 2

p,ν

�

ghR2,

tTp

. (3)

In the present paper we estimate the shape of f1 and f2 based on an experimental data set from the DeRisk project[2]. The data set covers a limited parameter set relevant for offshore wind turbine monopiles in storm conditions,and we therefore reduce the parameter space accordingly. First, the peak enhancement parameter γ is expressed asa function of Hs and Tp (see [9]). Next, since the tests were carried out with a fixed pile diameter, the influence ofR/gT 2

p is neglected. Since the sea states studied are all storm sea states, all the wave episodes considered are to goodapproximation in the long wave regime. Further, the Reynolds number dependence through ν is ignored along withany other scale effects from e.g. aeration. This leads to the simplified dependencies

P�

Fi

ρgHsR2 ≤F

ρgHsR2

�

= f1

Fi

ρghR2 ,Hs

gT 2p,

hgT 2

p

(4)

FFi= f2

Fi

ρghR2 ,Hs

gT 2p,

hgT 2

p,

tTa

. (5)

Here, the functional dependencies have further been modified such that the force levels are normalized with Hs insteadof h in the denominator to accommodate the expected linear behaviour and where time in f2 is normalized with a newtime scale Ta, which will be defined later in terms of γ and Tp. The role of the present work is thus to investigate thefunctional dependencies of ( f1, f2) to the wave parameters by analysis of experimental results. Further, the shape off2 is compared to the NewForce signal to investigate how well the NewForce model can reproduce the shape.

3. The NewForce model

The NewWave theory was derived by [1,11,17]. Given a power spectrum of free surface elevation S η, the NewWavetheory establishes the expected time history of free surface elevation around a peak as

ηNewWave =αη

σ2η

�

jRe�

S η(ω j)∆ω exp�

i�

ω j(t − t0) − k j(x − x0)���

, (6)

where αη is the crest height and σ2η = η

2 =� ∞

ω=0 S η(ω) dω. It is thus a linear theory and the associated linear forcehistory can be derived through its linear transfer function. Assuming that the wave force is inertia-dominated, thusneglecting the drag-contribution, this transfer function can be obtained through the Morison equation

Γ(ω) = iρπR2CMω2/k (7)

and involves a phase shift of π/2 through the i-factor. Further, CM is the inertia coefficient and k the wave number.The inline force for the NewWave free surface history (6) is thus

FNewWave =αη

σ2η

�

jRe�

S η(ω j)Γ(ω)∆ω exp�

i�

ω j(t − t0) − k j(x − x0)���

. (8)

The NewForce model takes basis in the linear force spectrum S F(ω) = |Γ(ω)|2S η(ω). We re-apply the principle ofNewWave to obtain the expected force history around a force peak from the linear force spectrum:

F(1)NewForce =

αF

σ2F

�

jRe�

�

�

�Γ(ω j)�

�

�

2 S η(ω j)∆ω exp�

i�

ω j(t − t0) − k j(x − x0)��

�

. (9)

226 Signe Schløer et al. / Energy Procedia 137 (2017) 223–2374 Author name / Energy Procedia 00 (2016) 000–000

Here αF is the force peak value and σ2F is the integrated force spectrum. Due to its linear origin, the NewForce signal

is symmetric around the peak. The associated time history of the free surface elevation is obtained by division withthe force transfer function Γ(ω) as

η(1)NewForce =

αF

σ2F

∑

jRe{

Γ∗(ω j)S η(ω j)∆ω exp(

i(

ω j(t − t0) − k j(x − x0)))}

, (10)

where Γ∗ is the complex conjugate of Γ.We note that the free surface elevation history associated with the NewForce model differs from the classical

NewWave theory by a phase shift of π/2 of all the wave components and multiplication by the linear force transferfunction inside the sum. The wave time history that produces a given force with largest probability is thus differentfrom a simple phase shift of π/2 of the classical NewWave signal. The NewForce signal principle can readily beapplied to other quantities of interest such as e.g. overturning moment as long as this can be expressed through alinear transfer function applied to the free surface elevation signal.

The NewForce signal is only first-order accurate, but the second order contribution can easily be added by 1)calculation of the second-order components of the incident wave field and kinematics by the the second order wavetheory of [16] and 2) by adding the second-order contributions from the slender-body force model that goes beyondlinear theory. In the analysis it is decided only to use Morison equation up to second order to be consistent with theused wave theory

F(2)NewForce =ρπR

2CM

0∫

−h

(u(2)t + u(1)u(1)

x + w(1)u(1)z )dz + ρRCD

0∫

−h

u(1)|u(1)|dz

+ρπR2(CM − 1)0∫

−h

(u(1)w(1)z )dz + ρπR2CMη

(1)NFu(1)

t .

(11)

One reason for this, besides mathematical strictness, is that we will apply fully nonlinear wave kinematics in futurework and therefore wish to work with a strictly linear, strictly second-order and fully nonlinear model. In (11) u andw are the horizontal and vertical particle velocities and ut =

∂u∂t the horizontal acceleration. Subscripts x and z means

that the variables are differentiated with respect to these coordinates. The nonlinear Rainey terms, [13]-[14], are alsoincluded to second order in (11), third term. For the drag and inertia coefficients the generic values CD = 1 andCM = 2 have been used troughout. It should be noted that while the first-order NewForce (and NewWave) models isthe most likely realization of a linear event of given peak amplitude, the second-order versions constructed by simpleadd-on of the second-order tems differ slightly from the most likely second-order event. For free surface elevation[10] derives expressions for the most likely second-order events while [6] calculate the most likely second-order forceand free surface events by a first-order reliability method.

4. The model tests

The model tests were conducted at DHI Denmark, as part of the DeRisk-project (2015-2019, [2]). The domainwas 18x20m and both uni-and multidirectional wave realizations based on JONSWAP-spectra were considered. Theexperiment was Froude-scaled with a scaling factor of 1:50.

Two water depths of h = 0.66m and h = 0.4m, 33 m and 20 m in full scale, with a flat sea bed were considered.The stiff pile had a diameter of D = 0.14m, 7 m in full scale, and was fixed at the bed and top. In the following, ifnothing else is stated, all data are given in full scale. The scale effects of the model tests which exist for the impactpressures, impact forces and viscous loads are therefore omitted in the present study. However, for inertia dominatedimpacts, the scale effects are usually considered small.

The free surface elevations considered in the analysis were measured 0.1m (lab scale) in front of the pile. Both theforces in the inline and transverse wave direction were measured by force transducers, however in this analysis onlythe inline force is considered.

Signe Schløer et al. / Energy Procedia 137 (2017) 223–237 227Author name / Energy Procedia 00 (2016) 000–000 5

The force transducers were excited when large waves hit the structure. To ensure that this excitation is not partof the analysed force-signal, the excitation was filtered out using a low pass filter. Considering the force spectrumof figure 1 the spectrum contains energy around 0.1 Hz due to the wave signal but the signal also contains peaks forfrequencies larger than 3 Hz, which is much higher than the wave frequencies. Frequencies higher than 2.5 Hz aretherefore disregarded in the analysis. Also in figure 1, a force signal before and after the filtering is seen. It is easy tosee that the force is smoothened after the filtering, but it is also seen that the slamming force (the peak of the force attime t ∼ 7760s) is maintained.

4.1. The sea states

In the present paper only the uni-directional waves are analyzed. Five sea states on a water depth of 33 m andsix sea states on a water depth of 20 m are considered. The lengths of the time series are 6 hours. The sea statesare presented in table 1, where the significant wave height, peak wave period and water depth are listed. Figure 2from [4] shows the steepness of the waves H/(gT 2) as function of the nondimensional water depth h/(gT 2). Thelarger H/(gT 2), the more steep the waves of the sea state. Similarly, smaller values of h/(gT 2), corresponds to moreshallow depth conditions. For the considered sea states the steepness and nondimensional water depths are based onthe significant wave height, Hs and peak wave period, Tp, also stated in the figure.

It is clear that none of the sea states can be described accurately with linear wave theory. Further the sea statesare grouped after their h/(gT 2

p)-value, indicated with black circles. Three groups are defined with the average valuesh/(gT 2

p) = 0.9 , 0.014 and 0.024. The color of each sea state is repeated through out the paper.

Table 1: Significant wave height, peak wave period and water depth of the considered wave realizations.

Test Hs Tp h(m) (s) (m)

9 7.5 12 3310 7.5 15 3311 9.5 12 3312 9.5 15. 3313 11.0 15 3320 5.8 12 2021 5.8 15 2022 6.8 12 2023 6.8 15 2024 7.5 15 2025 5.8 9 20

5. Results

5.1. Exceedance probability distributions of the free surface elevation and force signals

In the following the functional dependencies of f1 and f2, (4)-(5), are investigated. In figure 3 the exceedanceprobability curves of the crest height and force peaks for all wave realizations are shown. The crest heights and forcepeaks are the largest value in between two neighbouring zero-down-crossings of the free surface elevations, which aresorted in increasing order. The probability of exceedance is here calculated as

P = 1 − i−1N , i = 1, 2, 3, ...,N, (12)

where N is the number of waves in the time series. The crest heights and the force peaks are nondimenionalised asη/HS and F/(ρgHsR2).

The curves in figure 3 are not fully identical. As the probability of exceedance decreases the difference betweenthe curves increases. Furthermore, the relative difference between the smallest and largest crest height for the lowestexceedance probability is smaller than the difference between the smallest and largest force peaks. Thus, the expected

228 Signe Schløer et al. / Energy Procedia 137 (2017) 223–2376 Author name / Energy Procedia 00 (2016) 000–000

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

f (Hz)

10 -2

10 0

10 2

10 4

S F (kN

2s)

7750 7755 7760 7765 7770 7775

t (s)

-5000

0

5000

10000

F (k

N)

BeforeAfter

Fig. 1: The power spectrum and the time series of the measured forcesignal in full scale. Frequencies above 2.4 Hz are filtered out in the force-signal.

h/(gT2)H/(gT a2 )

0.001 0.02 0.2

10 4

10 3

10 2

Hs=, Tp=7.8m, 11.8s7.4 m, 12.6 s9.8 m, 11.8 s9.2 m, 15.1 s10.3 m, 15.1 s6.1 m, 12.2 s6.1 m, 12.6 s7.1 m, 12.4 s7.0 m, 14.7 s7.6 m, 14.7 s6.1 m, 8.9 s

Fig. 2: Figure from [4] used to classify the sea states, based on Hs andTp. � h = 33m, △ h = 20m. The black circles indicate the three groupsthe sea states are divided into as function of h/(gT 2

p).

linear behaviour of the crest heights and force peaks being proportional to the wave height does not hold for these seastates.

This means that the waves, as expected, are nonlinear and that their dependence to h/(gT 2p) cannot be ignored. For

the further analysis, we therefore change the normalisation to F/(ρghR2) and η/h to make it independent of Hs. Thisway, events of same wave height or force level can be picked from sea states of different significant wave height andbe compared. The associated exceedance probability curves are shown in figure 4. The difference between the curvesare larger relative to figure 3 where the linear force dependence to Hs was included in the normalization.

The closer the sea states are to the wave breaking limit, cf. figure 2, the larger the force-peaks are for sameprobability value since the shape of the surface elevations become more steep the closer the waves are to breakingresulting in larger forces.

For some of the exceedance probability curves of the crest heights the tangents of the curves for the smallestprobabilities is almost vertical, for example the sea state with Hs = 7.1m and Tp = 12.4s. If the largest crest height isleft out of account, the values of the five second largest crest heights are almost identical. It could look like a breakinglimit for which the crest height can not be larger. Comparing the time series of the force and free surface elevation,the force peaks corresponding to the largest five crest heights η/h are represented within the eight largest force values,F/(gρhR2). This indicates that the waves are or are close to breaking and that the forces are due to wave slamming.

As shown in (4) the probability distributions are functions of Hs/(gT 2p) and h/(gT 2

p) but the ordering in magnitudeis not coinciding. To illustrate this, nine peaks nearest different force values F/(ρghR2) are considered for all seastates in figure 5. Nine force peaks nearest F/(ρghR2) = 1.0 and F/(ρghR2) = 1.3 are indicated with red and blackasterisks, respectively. In the figure the corresponding crest heights for the same waves are also indicated with redand black asterisks. It is seen that there is large variation in these crest values, but also there is a tendency that thatthe variation is smaller for the sea states where the considered force values, F/(ρghR2) = 1.0 and F/(ρghR2) = 1.3,are near the extreme force values. This could indicate that for the extreme forces of a sea state, the number of shapeswhich can result in these forces are reduced. In figure 6 the relation between the wave height and period of the wavesresulting in these peaks are compared to the breaking criteria of [7]

Signe Schløer et al. / Energy Procedia 137 (2017) 223–237 229Author name / Energy Procedia 00 (2016) 000–000 7

0 0.2 0.4 0.6 0.8 1 1.2 1.4

10−3

10−2

10−1

100

η/Hs (−)

P (−

)

0 1 2 3 4 5 6 7 8

10−3

10−2

10−1

100

F/(ρgHsR2) (−)

P (−

)

Hs=, T

p=

7.8m, 11.8s7.4 m, 12.6 s9.8 m, 11.8 s9.2 m, 15.1 s10.3 m, 15.1 s6.1 m, 12.2 s6.1 m, 12.6 s7.1 m, 12.4 s7.0 m, 14.7 s7.6 m, 14.7 s6.1 m, 8.9 s

Fig. 3: Probability of exceedance of the crest heights and force peaks, when normalizing with HS . � h = 33m, △ h = 20m.

Hb

L0= A(

1 − exp(

−1.5πhL0

))

. (13)

Here L0 is the deep water wave length and Hb the limiting wave height. For irregular waves breaking occur for0.12 < A < 0.18. For linear waves, L0 = gT 2/(2π) and the nondimensional values Hb

L0and h

L0therefore correspond to

the nondimensional values H(gT 2) and h

(gT 2) on figure 2. A trend between the force values and the wave heights is seen.All the waves lie on a curve almost parallel to the breaking limits, and as the force value F/(gρhR2) increases, thewaves are closer to breaking.

5.2. The averaged force shape

In the following it is investigated if a similarity of the force shapes with same force value F/(ρghR2) but differentexceedance probability can be found. Irregular wave realizations are a stochastic process. In previous studies, wherethe NewWave is compared to extreme wave events in measured and calculated wave realizations, [8]-[19], the extremewaves are the average of a given number of the waves with largest crest height in the wave realization. The sameapproach is used in the present analysis, but on different levels of crest heights and force peak values.

In figure 7, the nine force shapes of the nine force peaks nearest F/(ρghR2) = 1.0 and F/(ρghR2) = 1.3 are shownfor the sea state with Hs = 10.3 m, Tp = 15.1 s and h = 33 m together with the average force time series, FA. Theforce peaks are centered at time t = 0 s. Before averaging the force shapes the forces are normalised with their peakvalues.

230 Signe Schløer et al. / Energy Procedia 137 (2017) 223–2378 Author name / Energy Procedia 00 (2016) 000–000

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

10−3

10−2

10−1

100

η/h (−)

P (−

)

0 0.5 1 1.5 2 2.5 3

10−3

10−2

10−1

100

F/(ρghR2) (−)

P (−

)

Hs=, T

p=

7.8m, 11.8s7.4 m, 12.6 s9.8 m, 11.8 s9.2 m, 15.1 s10.3 m, 15.1 s6.1 m, 12.2 s6.1 m, 12.6 s7.1 m, 12.4 s7.0 m, 14.7 s7.6 m, 14.7 s6.1 m, 8.9 s

Fig. 4: Probability of exceedance of the crest heights and force peaks, when normalizing with h. � h = 33m, △ h = 20m.

The standard deviations of the nine force shapes for each sea state are also shown. The variation is zero at t = 0and increases away from the force peaks. However, from the zero-upcrossing in front of the peak to the through afterthe peak the nine forces show a strong similarity.

In figure 8 the average force shapes of the force peaks with the force value F/(ρghR2) =(0.8, 1.0, 1.2, 1.3, 1.6) areshown for all sea states. The force shapes are grouped after the values of h/(gTp)2 of their corresponding sea states,as shown in figure 2. The time is normalised with the scale Ta, which is the time from the force peak to the troughafter the force peak of the linear NewForce signal for the same Tp-value. Thus, Ta is function of Tp and γ.

Below the force-shapes, the standard deviations of the average force shapes are shown. Generally, the shapes ofthe forces in each group match quite well and becomes more uniform as F/(ρghR2) increases.

The amplitude of the trough in front of the peak increases with increasing h/(gT 2p)-value and decreases with in-

creasing F/(ρghR2)-value. In the analysis it was tried to find a time-scale which would give all forces the samenondimensional period. However, as seen in the figures, we did not manage to find such a time scale, and the time,t/Ta, from the peak to the trough after the peak increases with increasing h/(gT 2

p). Still, we observe that the nondi-mensional time from the zero-upcrossing in front of the force peak to the through after the peak is almost identical forall forces for same h/(gT 2

p)-value when using this time normalization.The secondary load cycle is seen for the smallest force-values F/(ρghR2) ≥ 1.2. It could look like the secondary

load cycle force for h/(gT 2p) =0.009 occurs for smaller F/(ρghR2)-values compared to the other h/(gT 2

p)-values. Thisillustrates, that nonlinearity increases with decreasing water depth.

Considering the standard deviation, the deviation decreases as F/(ρghR2) increases. The standard deviation is zeroat t = 0 s, since all forces are normalized with their force peaks. The deviation increases on both sides of t = 0s toapproximately the time of occurrence of the troughs, showing that the average forces are most uniform around the

Signe Schløer et al. / Energy Procedia 137 (2017) 223–237 231Author name / Energy Procedia 00 (2016) 000–000 9

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5η/h (-)

10 -3

10 -2

10 -1

10 0

P (-

)

0 0.5 1 1.5 2 2.5 3

F/( ρghR2) (-)

10 -3

10 -2

10 -1

10 0

P (-

)

Fig. 5: Probability of exceedance of the crest heights and force peaks, when normalizing with h. � h = 33m, △ h = 20m. The red and blackasterisks shows the nine peak forces nearest the force values F/(gρhR2) = 1 and F/(gρhR2) = 1.3 and the corresponding crest values.

10−1 100 10110−2

10−1

H/L0(−)

h/L0(−)

A=0.12

A=0.18

(a) F/(gρhR2) = 0.8.

10 1 100 10110 2

10 1

h/L0( )

(b) F/(gρhR2) = 1.0.

10−1 100 10110−2

10−1

h/L0 (−)

(c) F/(gρhR2) = 1.3.

10−1 100 10110−2

10−1

h/L0 (−)

(d) F/(gρhR2) = 1.6.

Fig. 6: The breaking wave criteria of Goda [7]. The wave heights in the wave realizations corresponding to the 9 force peaks nearest the forcevalues F/(gρhR2) for all 11 sea states.

force-crest. This is in accordance with the theory of NewWave, [17], where it is shown that as the crest height ofthe NewWave increases the standard deviation remains constant meaning that the surface elevation becomes moredeterministic.

It is important to note that for smaller F/(ρghR2)-values the standard deviation increases. The above observationsindicating that the shapes of the normalized forces are a function of h/(gT 2

p), t/Ta and F/(ρghR2) are therefore onlygood approximations for roughly F/(ρghR2) > 0.8 in the present parameter range.

5.3. Comparison to the NewForce model

The NewForce model and the Morison equation are engineering tools, to calculate the wave forces. They aresuited for application, as they are simple and easily calculated. In this section it is investigated how well the different

232 Signe Schløer et al. / Energy Procedia 137 (2017) 223–23710 Author name / Energy Procedia 00 (2016) 000–000

−15 −10 −5 0 5 10 15−1

−0.5

0

0.5

1F/

F max

t (s)

F(t)/FmaxFAFA±σ

(a) F/(ρghR2) = 1.0.

−15 −10 −5 0 5 10 15−1

−0.5

0

0.5

1

F/F m

ax

t (s)

F(t)/FmaxFAFA±σ

(b) F/(ρghR2) = 1.3.

Fig. 7: The nine force shapes, F(t), and the corresponding average force, FA together with the standard deviation σ of the nine forces for the seastate with Hs = 6.8m, Tp = 15m and h = 20m.

force shapes presented in figure 8 can be predicted by the NewForce model extended to second order. The approachexplained in section 3 is used to calculate the second order NewForce signals, F(1)+(2)

NewForce = F(1)NewForce + F(2)

NewForce.Two examples with different F/(ρghR2)-value are shown in figure 9 for the sea state with Hs = 11m, Tp = 15s and

h = 33m. For comparison, the first and second order free surface elevations, ηNewForce based on the NewForce signalsare also shown.

Since extreme sea states are considered, some of the average force-shapes includes slamming, which is easy toidentify, as the force contains a ”hat” on top of the peak. This is seen in the average force in figure 9b. However, thewave slamming is not included in the NewForce model, and the slamming part of force-shape is therefore disregardedwhen compared to the NewForce signal. To account for this in the analysis, the average force shape around the crestis approximated by a fourth order polynomial, and the maximum value of this polynomial is instead used as targetpeak-value for the NewForce signal, α in (9).

Signe Schløer et al. / Energy Procedia 137 (2017) 223–237 233Author name / Energy Procedia 00 (2016) 000–000 11F/

(ρgh

R2 )=0.

8F/

(ρgh

R2 )=1.

0F/

(ρgh

R2 )=1.

2F/

(ρgh

R2 )=1.

3F/

(ρgh

R2 )=1.

6

-6 -4 -2 0 2 4 6-1

-0.5

0

0.5

1h/(gT p

2)=0.024

-6 -4 -2 0 2 4 6t/T a (-)

0

0.5

1

F/Fm

ax

-6 -4 -2 0 2 4 6-1

-0.5

0

0.5

1h/(gT p

2)=0.014

-6 -4 -2 0 2 4 6t/T a (-)

0

0.5

1-6 -4 -2 0 2 4 6

-1

-0.5

0

0.5

1

F/F

max

(-)

h/(gT p2)=0.009

-6 -4 -2 0 2 4 6t/T a (-)

0

0.5

1

σ(F

(t)/F

max

) (-)

-6 -4 -2 0 2 4 6-1

-0.5

0

0.5

1

-6 -4 -2 0 2 4 6t/T a (-)

0

0.5

1

F/Fm

ax

-6 -4 -2 0 2 4 6-1

-0.5

0

0.5

1

-6 -4 -2 0 2 4 6t/T a (-)

0

0.5

1-6 -4 -2 0 2 4 6

-1

-0.5

0

0.5

1

F/F

max

(-)

-6 -4 -2 0 2 4 6t/T a (-)

0

0.5

1

σ(F

(t)/F

max

) (-)

-6 -4 -2 0 2 4 6-1

-0.5

0

0.5

1

-6 -4 -2 0 2 4 6t/T a (-)

0

0.5

1

F/Fm

ax-6 -4 -2 0 2 4 6

-1

-0.5

0

0.5

1

-6 -4 -2 0 2 4 6t/T a (-)

0

0.5

1-6 -4 -2 0 2 4 6

-1

-0.5

0

0.5

1

F/F

max

(-)

-6 -4 -2 0 2 4 6t/T a (-)

0

0.5

1

σ(F

(t)/F

max

) (-)

-6 -4 -2 0 2 4 6-1

-0.5

0

0.5

1

-6 -4 -2 0 2 4 6t/T a (-)

0

0.5

1

F/Fm

ax

-6 -4 -2 0 2 4 6-1

-0.5

0

0.5

1

-6 -4 -2 0 2 4 6t/T a (-)

0

0.5

1-6 -4 -2 0 2 4 6

-1

-0.5

0

0.5

1

F/F

max

(-)

-6 -4 -2 0 2 4 6t/T a (-)

0

0.5

1

σ(F

(t)/F

max

) (-)

-6 -4 -2 0 2 4 6-1

-0.5

0

0.5

1

-6 -4 -2 0 2 4 6t/T a (-)

0

0.5

1-6 -4 -2 0 2 4 6

-1

-0.5

0

0.5

1

-6 -4 -2 0 2 4 6t/T a (-)

0

0.5

1-6 -4 -2 0 2 4 6

-1

-0.5

0

0.5

1

F/F

max

(-)

-6 -4 -2 0 2 4 6t/T a (-)

0

0.5

1

σ(F

(t)/F

max

) (-)

Fig. 8: The average force shape as function of h/(gT 2p) and F/(ρghR2. The color of each sea state corresponds to the colors in figure 2.

234 Signe Schløer et al. / Energy Procedia 137 (2017) 223–23712 Author name / Energy Procedia 00 (2016) 000–000

−6 −4 −2 0 2 4 6−1

−0.5

0

0.5

1η/

η max

t/Ta

ηNF(1)

ηNF(1)+(2)

−6 −4 −2 0 2 4 6−1

−0.5

0

0.5

1

t/Ta

F/F m

ax

FVal=3972kN

FAverageFAverage±σ

FNF(1)

FNF(1)+FM

(2)

(a) F/(ρghR2) = 1.0.

−6 −4 −2 0 2 4 6−1

−0.5

0

0.5

1

η/η m

ax

t/Ta

ηNF(1)

ηNF(1)+(2)

−6 −4 −2 0 2 4 6−1

−0.5

0

0.5

1

t/Ta

F/F m

ax

FVal=5277kN

FAverageFAverage±σ

FNF(1)

FNF(1)+FM

(2)

(b) F/(ρghR2) = 1.3.

Fig. 9: The average force for the sea state with Hs = 10.3m, Tp = 15.1s, h = 33m and h/(gT 2p) = 0.014 for two force values, and the corresponding

NewForce signal and surface elevation based on the NewForce model.

Comparing the first and second order NewForce signals, the second order NewForce signal compares better tothe average force signal. The trough in front of the peak is lifted upward and the force-crest becomes more narrow.This supports the finding that the time scale of the local force peak is not linear. It is therefore only the second orderNewForce signal which is used in the following analysis. It is further seen that the NewForce signal and the averageforce compares best for the smallest force peak value, F/(ρghR2) = 1.0, where the NewForce signal is inside thegrey area indicating F ± σ, with σ being the standard deviation. For F/(ρghR2) = 1.3 the NewForce signal does notcapture the asymmetry of the force shape, i.e. the force is more steep after the peak compared to in front of the peak.Higher-order terms are necessary to include such nonlinear effects.

The average force shapes of figure 8 are repeated in figure 10, but now they are compared to the correspondingsecond order NewForce signals. The NewForce signals for each h/(gTp) and F/(ρghR2) value are very similar andlie on top of each other for some situations. For the smallest force value, F/(ρghR2) = 0.8, the NewForce signalscompare very well to the average force shapes. However, as the force value increases, the differences between theaverage force shape and the NewForce signal increases. The dissimilarities are largest for the wave realizations inmost shallow water, h/(gT 2

p) = 0.009 and occur for smaller force values compared to the two other groups of h/(gT 2p).

This is a consequence of the stronger nonlinearity in shallow water, where accuracy of a second order model mustdecrease. The more shallow the more asymmetric the wave shapes become. The NewForce signal is more symmetric

Signe Schløer et al. / Energy Procedia 137 (2017) 223–237 235Author name / Energy Procedia 00 (2016) 000–000 13

around its crest value. This also explains why the crest of the NewForce signal is more steep in front of the peakcompared to the average force, while after the peak, the average force is more steep.

The comparison indicates that it is possible to define limits based on h/(gTp) and F/(ρghR2), for which the expectedforce shape for a given sea state can be represented by the NewForce model extended to second order.

F/(ρ

ghR2 )=

0.8

F/(ρ

ghR2 )=

1.0

F/(ρ

ghR2 )=

1.2

F/(ρ

ghR2 )=

1.3

F/(ρ

ghR2 )=

1.6

−5 0 5−1

−0.5

0

0.5

1

h/(gTp2)=0.009

F/F m

ax (−

)

t/Ta (−)

−5 0 5−1

−0.5

0

0.5

1

h/(gTp2)=0.014

t/Ta (−)

−5 0 5−1

−0.5

0

0.5

1

h/(gTp2)=0.024

t/Ta (−)

−5 0 5−1

−0.5

0

0.5

1

F/F m

ax (−

)

t/Ta (−)

−5 0 5−1

−0.5

0

0.5

1

t/Ta (−)

−5 0 5−1

−0.5

0

0.5

1

t/Ta (−)

−5 0 5−1

−0.5

0

0.5

1

F/F m

ax (−

)

t/Ta (−)

−5 0 5−1

−0.5

0

0.5

1

t/Ta (−)

−5 0 5−1

−0.5

0

0.5

1

t/Ta (−)

−5 0 5−1

−0.5

0

0.5

1

F/F m

ax (−

)

t/Ta (−)

−5 0 5−1

−0.5

0

0.5

1

t/Ta (−)

−5 0 5−1

−0.5

0

0.5

1

t/Ta (−)

−5 0 5−1

−0.5

0

0.5

1

F/F m

ax (−

)

t/Ta (−)

−5 0 5−1

−0.5

0

0.5

1

t/Ta (−)

−5 0 5−1

−0.5

0

0.5

1

t/Ta (−)

Fig. 10: The average force shape (full line) and the corresponding second force NewForce (dasped line) as function of h/(gT 2p) and

F/(ρghR2. The color of each sea state corresponds to the colors in figure 2.

236 Signe Schløer et al. / Energy Procedia 137 (2017) 223–237

14 Author name / Energy Procedia 00 (2016) 000–000

6. Summary and discussion

Analysis of extreme wave forces based on experiments with a stiff pile has been presented. As the force-peaks arenot only function of the wave height but also depends on e.g. the wave front slope, it is associated with uncertaintiesto base the design load on a design wave height only. The outset of the analysis was therefore to investigate if theforce time history of extreme wave forces could be predicted directly based on force statistics, without having anyinformation of the wave height and wave period resulting the specific force-value.

In the paper the NewForce model has been presented for the first time. In the NewForce model the NewWavetheory principle is re-applied to obtain the expected force history around a force peak from the linear force spectrum.

The functional dependencies of the exceedance probability curves of the measured inline force and the normalisedforce shapes was investigated. It was found that the probability distributions of the force peaks are functions of bothHs/(gT 2

p) and h/(gT 2p). For a given force value, the corresponding waves group around a curve parallel to the breaking

limits of [7].The normalised force shape was found to be function of F/(ρghR2), h/(gT 2

p) and t/Ta with good similarity forF/(ρghR2) > 0.8. Further, the second order NewForce model proved to predict the force shapes of moderate nonlinearwaves well. The performance was found to be best for the larger values of h/(gT 2

a ), agreeing with the general decreaseof nonlinearity towards larger depth. Thus, based on h/(gTp) and F/(ρghR2), limits can be defined, for which theexpected force shape for a given sea state can be represented by second order NewForce model.

The presented analysis demonstrates that it is possible to estimate the exceedance probability curves of the forcepeaks and the associated force shapes from the normalised sea state parameters Hs/(gT 2

p), h/(gT 2p), and that the

second-order NewForce model provides a good estimate of the expected force shape for waves of moderate non-linearity. The extension to multidirectional wave realizations is straightforward and should be investigated in futurework. While the present work addresses the linear expected event and adds the second-order terms, a direct determina-tion of the most likely second-order event for a given force can be obtained by application of a First-Order ReliabilityMethod (FORM). Such an investigation is presented by Ghadirian et al. [6] with comparison to the same data set andtreatment of directional waves.

Slamming loads could be observed for the largest force levels as a distinct ’hat’ on top of the smoother underlyingforce curve. While these seem beyond a simple slender-body force model, improved prediction of the force shapes ofmore nonlinear waves may be obtained by incorporating a wave model more accurate than second order. This couldbe the fully nonlinear wave model, OceanWave3D, [5] and is planned for future work.

Acknowledgment

DeRisk is funded by a research project grant from Innovation Fund Denmark, grant number 4106-00038B. Furtherfunding is provided by Statoil and the participating partners. All funding is gratefully acknowledged.

DHI Denmark is thanked for providing the experimental data as part of the DeRisk-project. Simulating discussionswith Prof. Paul Taylor, Dr. Thomas Adcock and Dr. Dripta Sarkar, University of Oxford, are acknowledged.

Signe Schløer et al. / Energy Procedia 137 (2017) 223–237 237Author name / Energy Procedia 00 (2016) 000–000 15

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